Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 5 / e

Normalize $Y_0=1/X_0$. Solving its stochastic differential equation givesFor logarithmic utility, $\hat{U}(y)=-\log y-1$. Part c and $\mathbb{E}{W}_T=0$ therefore giveIf ${\pi }_t=X_t\lambda /(S_t\sigma )$, part d gives $d(XY)=0$. Hence $X_t{Y}_t=X_0{Y}_0=1$, so $U^{\prime }(X_T)=1/X_T=Y_T$ and both inequalities in part c are equalities.